Heatmap, Flow Coefficient, Local Entropy, Weighted h-index and Redundancy
Source:R/centrality-batch9.R
centrality_heatmap.RdFive local measures. Heatmap centrality (Duron 2020) is the farness of a node minus the mean farness of its neighbors, \(C(v) = f(v) - \frac{1}{k_v} \sum_{u \in N(v)} f(u)\), with \(f\) the sum of hop distances to the reachable nodes. The flow coefficient (Honey et al. 2007) is the fraction of ordered pairs of distinct neighbors joined by a two-step path through the node and by no direct link, as in the Brain Connectivity Toolbox. Local entropy (Nie et al. 2016) is \(-\sum_{j \in N(i)} k_j \ln k_j\). The weighted h-index (Gao et al. 2019) is the h-index of the multiset in which each neighbor \(j\) contributes the value \(k_i k_j\) repeated \(k_j\) times. Redundancy (Burt 1992; Borgatti 1997) is the mean degree of the neighbors of a node within its ego network.
Usage
centrality_heatmap(x, mode = "all", ...)
centrality_flow_coefficient(x, ...)
centrality_local_entropy(x, mode = "all", ...)
centrality_weighted_h_index(x, mode = "all", ...)
centrality_redundancy(x, ...)Arguments
- x
Network input accepted by
centrality.- mode
For directed networks:
"all"(default),"out"or"in".- ...
Further arguments to
centrality, such asnormalized.
Value
A named numeric vector with one score per node, in input node order. The weighted h-index is an integer vector.
Details
Edge weights are ignored by all five. Heatmap, local entropy and the
weighted h-index follow mode. Redundancy ignores direction, and
the flow coefficient uses the direction of the links. On an undirected
network the flow coefficient of a node with at least two neighbors equals
one minus its clustering coefficient, and redundancy equals degree minus
effective size. Lower values mark more
central nodes for heatmap and local entropy. Isolated nodes return
NaN for heatmap and 0 for local entropy, and nodes with fewer than
two neighbors score 0 on the flow coefficient. The formula for local
entropy follows the Centrality Zoo and the survey of Omar and Plapper
(2021), which agree.
References
Duron, C. (2020). Heatmap centrality: A new measure to identify super- spreader nodes in scale-free networks. PLOS ONE, 15(7), e0235690.
Honey, C. J., Kotter, R., Breakspear, M., & Sporns, O. (2007). Network structure of cerebral cortex shapes functional connectivity on multiple time scales. PNAS, 104(24), 10240-10245.
Nie, T., Guo, Z., Zhao, K., & Lu, Z.-M. (2016). Using mapping entropy to identify node centrality in complex networks. Physica A, 453, 290-297.
Gao, L., Yu, S., Li, M., Shen, Z., & Gao, Z. (2019). Weighted h-index for identifying influential spreaders. Symmetry, 11(10), 1263.
Borgatti, S. P. (1997). Structural holes: Unpacking Burt's redundancy measures. Connections, 20(1), 35-38.
Examples
centrality_heatmap(regulation_net)
#> Explore Plan Monitor Adapt Reflect Discuss Synthesize
#> 0.4000000 -0.6666667 -1.7142857 -0.8333333 0.2000000 0.6000000 2.0000000
#> Evaluate Create Share
#> 1.0000000 -0.5000000 1.0000000
centrality_flow_coefficient(regulation_net)
#> Explore Plan Monitor Adapt Reflect Discuss Synthesize
#> 0.2500000 0.2333333 0.1904762 0.3000000 0.2000000 0.2000000 0.2500000
#> Evaluate Create Share
#> 0.2000000 0.2333333 0.3000000
centrality_local_entropy(regulation_net)
#> Explore Plan Monitor Adapt Reflect Discuss Synthesize
#> -45.64268 -54.05867 -61.93842 -51.35531 -43.30812 -48.34605 -43.16967
#> Evaluate Create Share
#> -53.92023 -56.56069 -53.92023
centrality_weighted_h_index(regulation_net)
#> Explore Plan Monitor Adapt Reflect Discuss Synthesize
#> 25 28 33 27 22 25 20
#> Evaluate Create Share
#> 25 30 25
centrality_redundancy(regulation_net)
#> Explore Plan Monitor Adapt Reflect Discuss Synthesize
#> 2.000000 2.333333 3.142857 1.666667 1.200000 1.600000 1.500000
#> Evaluate Create Share
#> 2.000000 2.666667 2.400000